Splitting a secret message in a way that a single person is not able to reconstruct it is a common task in information processing, especially for high-security applications. One solution to this problem and its generalization including several variations is provided by classical cryptography and is called “secret sharing.” Classical secret sharing involves splitting a message using mathematical algorithms and distributing the pieces to two or more legitimate users via classical communication. However, all classical communication systems are susceptible to eavesdropping attacks, which makes the secret vulnerable to unauthorized disclosure to unintended recipients.
Quantum cryptography systems exploit the quantum nature of particles to create an unconditionally secure communication system. Quantum cryptography systems employ the exchange of quantum particles, such as photons, that are encoded to form “qubits.” The use of such quantum particles has been proposed for quantum communication, such as for “quantum secret sharing”. Quantum communication can reveal eavesdropping in essentially the same manner as quantum cryptography, thus guaranteeing secure distribution of shared information.
The basic principles behind communicating between two parties with entangled qubits generated by a third party are understood as follows. Suppose we have two qubits prepared in one of two quantum states, |Ψ0 or |Ψ1. We now give one qubit to one party called “Alice” and one qubit to another party called “Bob.” Both parties know that the state of the qubit is either |Ψ0 or |Ψ1, and their task is to perform local measurements on their qubits and communicate through a classical channel to determine the state they have been given. Alice and Bob can perfectly distinguish between the states using local operations and classical communication only if the states are orthogonal. When |Ψ0 and |Ψ1 are not orthogonal, Alice and Bob can use two different approaches to distinguish between the states.
The first approach is the minimum error state discrimination approach. In this approach, after Alice and Bob measure their qubits, each has to give a conclusive answer about the measured state and they are not allowed to give “I don't know” as an answer. However, since the states are not orthogonal, the price that the two parties must pay for giving a definite answer is the chance that they will make a mistake and incorrectly identify the state. The minimum probability of making a wrong guess, when each state is equally likely, is
                              p          E                =                              1            2                    ⁢                                    (                              1                -                                                      1                    -                                                                                                                    〈                                                                                    Ψ                              0                                                        |                                                          Ψ                              1                                                                                〉                                                                                            2                                                                                  )                        .                                              (        1        )            
An alternative approach to the state discrimination problem is the unambiguous state discrimination approach. In this approach, some measurement outcomes are allowed to be inconclusive, so that Alice and Bob might fail to identify the state. However, if they succeed they will not make an error. If each state is equally likely and both qubits are measured together, then the optimal probability to successfully and unambiguously distinguish the states is given by:pidp=1−|Ψ0|Ψ1|.  (2)
The probability of getting an inconclusive result, which provides no information about the state, is 1−pidp. This success probability can also be achieved if Alice and Bob measure the qubits separately, and if they are allowed to communicate the measurement result through a classical channel. In this procedure, Alice makes a projective measurement on her qubit that gives her no information about the state, and she then communicates the result of her measurement to Bob. Based on this information, Bob is able to make a measurement on his qubit that allows him to decide, with a success probability of pidp, what the initial state was.
There are difficulties if one wants to use this procedure as part of a quantum communication scheme. In a secret sharing scheme based on the above, Alice and Bob are each recipients of different parts of a message or key sent by a third-party “sender” named “Charlie.” The message parts need to be combined in order for the message or key to be revealed. The first problem, then, is that if the parts are to be combined at a time significantly later than when they were sent, quantum memory is required to store the qubit states, i.e., the qubits have to be protected against decoherence for a long time.
If one attempts to surmount this difficulty by having the parties measure their qubits immediately upon receiving them, one is faced with the problem that the information gain is asymmetric. Alice learns nothing about the key, and Bob learns everything. The only way this could be useful is if Alice and Bob are in the same location and are to use the key immediately. If they are in separate locations and will be using the key later, another procedure is required.
In one approach, both parties measure their qubit immediately upon receiving it, each obtaining a result of either 0 or 1. There are thus four sets of possible measurement results: {0,0}, {1,1}, {0,1}, and {1,0}. The result {0,0} corresponds to |Ψ0, the result {1,1} corresponds to |Ψ1 and the results {0,1}, and {1,0} correspond to a “failure.” It has been shown that when the two states have the same Schmidt basis, the probability of successfully identifying the state is given pidp. This approach can be used in a secret-sharing scheme, wherein the set of measurement results obtained by Alice and Bob, which is classical information, can be stored indefinitely and compared at a later time to reveal the key.
This approach, however, has a significant drawback. The key bits for which the measurement failed, and which, therefore, must be discarded, are only identified after Alice and Bob have compared their bit strings. It would be much more efficient if the bits that must be discarded could be identified immediately.